The success of methods based on reversed unitarity provides an explanation of the scarcity
of improvements regarding angular integrals. However, the direct approach towards angular
integrals, rather than treating all PSIs by reversed unitarity still has its merits for certain
applications. It is conceptually simple, since it makes straightforward use of their properties
rather than taking a detour to loop integrals. One might compare the calculation of the massless Neerven integral
in Ref. [1]} to ours in Sec. REF  as an example. By considering a specific class of PSIs,
in the present paper angular integrals with two denominators, analytic results can be obtained
in closed form and expanded to all orders in \(\varepsilon \) . Furthermore, recursion relations
can explicitly be solved. This yields a completely solved building block which can be plugged
into calculations, without the need to run an IBP reduction and to solve differential equations
order by order. The expansion presented in Sec. REF  has the benefit
to be compatible with commonly used methods since the result is expressed in terms of multiple
polylogarithms. Of course, employing the angular integral discussed here is only applicable for
observables with phase space integrals of appropriate form and is not simply generalizable for
other PSIs. However, the ideas presented should also be of value for related PSIs like angular integrals with 3 and
more denominators. Such integrals appear, e.g., as master integrals in the reversed unitarity
method (see Ref. [2]}).
